Question

Solve the differential equation: $\frac{dy}{dx}+y\tan x=\sin x$

Answer 1

We have, $\frac{dy}{dx}+y\tan x=\sin x$

Comparing with, standard first order linear differential equation

$\frac{dy}{dx}+Py=Q$

We get $P=\tan x$ and $Q=\sin x$

Thus, integrating factor. $\text{I.F}=e^{\int Pdx}=e^{\int \tan xdx}=e^{\ln \sec x}=\sec x$

Therefore solution is given by,

$y\left(\text{I.F.}\right)=\int Q\left(\text{I.F}\right)dx+C$

$\Rightarrow y\left(\sec x\right)=\int \sec x\sin xdx$

$\Rightarrow \left(\sec x\right)y=\int \tan xdx+C$

$\Rightarrow \left(\sec x\right)y=\ln |\sec x|+C$

$\Rightarrow y=\cos x\ln |\sec x|+C\cos x$